Editing Localization of a category (section)

===Model categories===
A rigorous construction of localization of categories, avoiding these set-theoretic issues, was one of the initial reasons for the development of the theory of [[Model category|model categories]]: a model category ''M'' is a category in which there are three classes of maps; one of these classes is the class of [[weak equivalence (homotopy theory)|weak equivalence]]s. The [[homotopy category]] Ho(''M'') is then the localization with respect to the weak equivalences. The axioms of a model category ensure that this localization can be defined without set-theoretical difficulties.